Problem

Source: China South East Mathematical Olympiad Grade 10 Prob. 4

Tags: combinatorics



For any four points on a plane, if the areas of four triangles formed are different positive integer and six distances between those four points are also six different positive integers, then the convex closure of $4$ points is called a "lotus design." (1) Construct an example of "lotus design". Also what are areas and distances in your example? (2) Prove that there are infinitely many "lotus design" which are not similar.